Number of four letter words can be formed using the letters of word VIBRANT if letter V is must included, are :

Options

A. 840
B. 480
C. 120
D. 240

Answer

To solve the problem of finding the number of four-letter words that can be formed using the letters of the word "VIBRANT" with the condition that the letter 'V' must be included, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Letters**: The word "VIBRANT" consists of 8 distinct letters: V, I, B, R, A, N, T.

2. **Include the Letter 'V'**: Since 'V' must be included in every four-letter word, we can consider 'V' as one of the letters in our word.

3. **Choose Remaining Letters**: Since we need a total of 4 letters and 'V' is already included, we need to select 3 more letters from the remaining letters: I, B, R, A, N, T. This gives us a total of 7 letters (I, B, R, A, N, T) to choose from.

4. **Calculate the Combinations**: We need to choose 3 letters from these 7 remaining letters. The number of ways to choose 3 letters from 7 is given by the combination formula:
\[ \text{Number of ways} = \binom{7}{3} \]

5. **Calculate the Value of Combinations**:
\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

6. **Arrange the Letters**: Now, we have 4 letters in total (V and the 3 chosen letters). The number of ways to arrange these 4 letters is given by:
\[ 4! = 24 \]

7. **Total Number of Words**: Finally, to find the total number of four-letter words that can be formed, we multiply the number of ways to choose the letters by the number of ways to arrange them:
\[ \text{Total Words} = \binom{7}{3} \times 4! = 35 \times 24 = 840 \]

### Final Answer:
The total number of four-letter words that can be formed using the letters of the word "VIBRANT" with 'V' included is **840**.